Subgroups — C₁₉×#19054 #18025013

Order 4864.

926 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#3235	C₂	1	2	yes	maximal, complemented
3	C₁₉×#3150	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1660	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1657	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1536	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1531	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₉×#135	—	2	2	no	—
9	C₁₉×#563	C₂²	1	4	yes	—
10	C₁₉×#559	—	2	1	no	—
11	Q₈²×C₁₉	—	2	1	no	—
12	C₁₉×#556	—	2	2	no	—
13	D₄×Q₈×C₁₉	—	2	2	no	—
14	C₁₉×#542	—	2	4	no	—
15	C₁₉×#536	—	2	2	no	—
16	C₁₉×#533	C₂²	1	2	yes	—
17	C₁₉×#532	C₂²	1	1	yes	characteristic
18	C₁₉×#532	C₂²	1	1	yes	characteristic
19	C₁₉×#501	C₂²	1	4	yes	—
20	C₁₉×#497	—	2	1	no	—
21	C₁₉×#497	C₂²	1	2	yes	—
22	C₁₉×#496	—	2	1	no	—
23	C₁₉×#496	C₂²	1	2	yes	—
24	C₁₉×#490	C₂²	1	4	yes	—
25	C₁₉×#482	—	2	4	no	—
26	C₁₉×#476	—	2	2	no	—
27	C₁₉×#474	—	2	2	no	—
28	C₁₉×#432	C₂²	1	2	yes	—
29	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
30	C₁₉×#330	C₂²	1	8	yes	—
31	C₁₉×#329	C₂²	1	4	yes	—
32	C₂×#41×C₁₉	—	2	1	no	—
33	C₂×#41×C₁₉	D₄	1	2	yes	—
34	C₂²×Q₈×C₁₉	—	2	2	no	—
35	C₁₉×#135	—	4	8	no	—
36	C₂×Q₁₆×C₁₉	—	2	2	no	—
37	C₂×SD₁₆×C₁₉	—	2	2	no	—
38	C₂×SD₁₆×C₁₉	—	2	2	no	—
39	C₂×D₈×C₁₉	—	2	2	no	—
40	C₂²×C₈×C₁₉	—	2	1	no	—
41	C₁₉×C₄⋊Q₈	—	2	4	no	—
42	C₁₉×C₄⋊Q₈	—	2	4	no	—
43	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
45	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
46	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
47	C₁₉×#127	D₄	1	1	yes	characteristic
48	C₁₉×#127	C₂³	1	1	yes	characteristic
49	C₁₉×#126	—	2	8	no	—
50	C₁₉×#125	C₂³	1	4	yes	—
51	C₁₉×#124	—	2	4	no	—
52	C₁₉×#124	—	2	4	no	—
53	C₁₉×#124	—	2	1	no	—
54	C₁₉×#124	C₂³	1	2	yes	—
55	C₁₉×#123	—	2	4	no	—
56	C₁₉×#123	—	2	4	no	—
57	C₁₉×C₂²⋊Q₈	—	2	4	no	—
58	C₁₉×C₂²⋊Q₈	—	2	2	no	—
59	C₁₉×C₂²⋊Q₈	—	2	2	no	—
60	C₁₉×C₂²⋊Q₈	—	2	8	no	—
61	C₁₉×#121	—	2	2	no	—
62	C₁₉×#121	—	2	2	no	—
63	C₄×Q₈×C₁₉	—	2	4	no	—
64	C₄×Q₈×C₁₉	—	2	2	no	—
65	C₄×Q₈×C₁₉	—	2	2	no	—
66	C₄×D₄×C₁₉	—	2	4	no	—
67	C₄×D₄×C₁₉	—	2	2	no	—
68	C₄×D₄×C₁₉	—	2	2	no	—
69	C₁₉×#117	—	2	2	no	—
70	C₁₉×#117	D₄	1	4	yes	—
71	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
72	C₁₉×#107	—	2	4	no	—
73	C₁₉×#106	—	2	4	no	—
74	C₁₉×C₄⋊C₈	—	2	2	no	—
75	C₁₉×#102	—	2	4	no	—
76	C₁₉×#102	—	2	4	no	—
77	C₁₉×C₂²⋊C₈	—	2	2	no	—
78	C₁₉×#97	—	2	1	no	—
79	C₁₉×#97	C₂³	1	2	yes	—
80	C₁₉×#97	C₂³	1	4	yes	—
81	#41×C₁₉	—	4	4	no	—
82	#41×C₁₉	—	4	4	no	—
83	C₂×Q₈×C₁₉	—	2	4	no	—
84	C₂×Q₈×C₁₉	—	2	2	no	—
85	C₂×Q₈×C₁₉	—	2	2	no	—
86	C₂×Q₈×C₁₉	—	2	1	no	—
87	C₂×Q₈×C₁₉	—	8	4	no	—
88	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
89	C₂×D₄×C₁₉	—	2	2	no	—
90	C₂×D₄×C₁₉	—	2	2	no	—
91	C₂×D₄×C₁₉	—	2	2	no	—
92	C₂×D₄×C₁₉	—	2	1	no	—
93	C₂×D₄×C₁₉	—	4	1	no	—
94	C₂×D₄×C₁₉	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₁₉	—	2	4	no	—
96	C₂²×C₄×C₁₉	—	2	2	no	—
97	C₂²×C₄×C₁₉	—	4	1	no	—
98	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
100	Q₁₆×C₁₉	—	4	4	no	—
101	SD₁₆×C₁₉	—	4	4	no	—
102	SD₁₆×C₁₉	—	4	4	no	—
103	D₈×C₁₉	—	4	4	no	—
104	C₂×C₈×C₁₉	—	2	4	no	—
105	C₂×C₈×C₁₉	—	2	2	no	—
106	C₂×C₈×C₁₉	—	2	4	no	—
107	C₄⋊C₄×C₁₉	—	2	8	no	—
108	C₄⋊C₄×C₁₉	—	2	2	no	—
109	C₄⋊C₄×C₁₉	—	2	2	no	—
110	C₄⋊C₄×C₁₉	—	2	2	no	—
111	C₄⋊C₄×C₁₉	—	2	4	no	—
112	C₄⋊C₄×C₁₉	—	2	4	no	—
113	C₄⋊C₄×C₁₉	—	2	2	no	—
114	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
115	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
116	C₂²⋊C₄×C₁₉	—	2	4	no	—
117	C₂²⋊C₄×C₁₉	—	2	8	no	—
118	C₂²⋊C₄×C₁₉	—	2	4	no	—
119	C₂²⋊C₄×C₁₉	—	2	4	no	—
120	C₂²⋊C₄×C₁₉	—	4	2	no	—
121	C₄²×C₁₉	—	2	4	no	—
122	C₄²×C₁₉	—	2	4	no	—
123	C₄²×C₁₉	C₂×D₄	1	2	yes	—
124	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
126	#19054	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₁₉	—	2	2	no	—
128	C₂³×C₁₉	—	4	1	no	—
129	C₂³×C₁₉	#120	1	1	yes	characteristic
130	Q₈×C₁₉	—	4	2	no	—
131	Q₈×C₁₉	—	4	4	no	—
132	Q₈×C₁₉	—	4	2	no	—
133	Q₈×C₁₉	—	4	4	no	—
134	Q₈×C₁₉	—	8	4	no	—
135	D₄×C₁₉	—	4	2	no	—
136	D₄×C₁₉	—	4	4	no	—
137	D₄×C₁₉	—	4	2	no	—
138	D₄×C₁₉	—	4	4	no	—
139	D₄×C₁₉	—	8	2	no	—
140	D₄×C₁₉	—	8	2	no	—
141	C₂×C₄×C₁₉	—	2	2	no	—
142	C₂×C₄×C₁₉	—	2	4	no	—
143	C₂×C₄×C₁₉	—	2	2	no	—
144	C₂×C₄×C₁₉	—	2	4	no	—
145	C₂×C₄×C₁₉	—	4	2	no	—
146	C₂×C₄×C₁₉	—	4	4	no	—
147	C₂×C₄×C₁₉	—	4	1	no	—
148	C₂×C₄×C₁₉	—	8	4	no	—
149	C₂×C₄×C₁₉	—	8	2	no	—
150	C₂×C₄×C₁₉	#120	1	2	yes	—
151	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
152	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
153	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
154	C₈×C₁₉	—	2	4	no	—
155	C₈×C₁₉	—	4	4	no	—
156	#3550	C₂×C₁₉	1	1	yes	characteristic, complemented
157	#3235	C₂×C₁₉	1	2	yes	complemented
158	#3150	C₂×C₁₉	1	1	yes	characteristic
159	#1660	C₂×C₁₉	1	4	yes	complemented
160	#1657	C₂×C₁₉	1	4	yes	complemented
161	#1536	C₂×C₁₉	1	1	yes	characteristic, complemented
162	#1531	C₂×C₁₉	1	2	yes	complemented
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	2	no	—
165	C₂²×C₁₉	—	4	2	no	—
166	C₂²×C₁₉	—	4	4	no	—
167	C₂²×C₁₉	—	8	1	no	—
168	C₂²×C₁₉	—	8	2	no	—
169	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	2	no	—
172	C₄×C₁₉	—	2	2	no	—
173	C₄×C₁₉	—	4	2	no	—
174	C₄×C₁₉	—	4	4	no	—
175	C₄×C₁₉	—	4	4	no	—
176	C₄×C₁₉	—	8	1	no	—
177	C₄×C₁₉	#453	1	2	yes	—
178	C₂×#135	—	2	2	no	—
179	#563	C₂²×C₁₉	1	4	yes	—
180	#559	—	2	1	no	—
181	Q₈²	—	2	1	no	—
182	#556	—	2	2	no	—
183	D₄×Q₈	—	2	2	no	—
184	#542	—	2	4	no	—
185	#536	—	2	2	no	—
186	#533	C₂²×C₁₉	1	2	yes	—
187	#532	C₂²×C₁₉	1	1	yes	characteristic
188	#532	C₂²×C₁₉	1	1	yes	characteristic
189	#501	C₂²×C₁₉	1	4	yes	—
190	#497	—	2	1	no	—
191	#497	C₂²×C₁₉	1	2	yes	—
192	#496	—	2	1	no	—
193	#496	C₂²×C₁₉	1	2	yes	—
194	#490	C₂²×C₁₉	1	4	yes	—
195	#482	—	2	4	no	—
196	#476	—	2	2	no	—
197	#474	—	2	2	no	—
198	#432	C₂²×C₁₉	1	2	yes	—
199	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
200	#330	C₂²×C₁₉	1	8	yes	—
201	#329	C₂²×C₁₉	1	4	yes	—
202	C₂×C₁₉	—	4	1	no	—
203	C₂×C₁₉	—	8	2	no	—
204	C₂×C₁₉	—	16	1	no	—
205	C₂×C₁₉	C₂×#453	1	1	yes	characteristic
206	C₂×C₁₉	#3020	1	2	yes	—
207	C₂×#41	—	2	1	no	—
208	C₂×#41	D₄×C₁₉	1	2	yes	—
209	C₂²×Q₈	—	2	2	no	—
210	#135	—	4	8	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×SD₁₆	—	2	2	no	—
214	C₂×D₈	—	2	2	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	4	no	—
217	C₄⋊Q₈	—	2	4	no	—
218	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
219	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
220	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
221	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
222	#127	D₄×C₁₉	1	1	yes	characteristic
223	#127	C₂³×C₁₉	1	1	yes	characteristic
224	#126	—	2	8	no	—
225	#125	C₂³×C₁₉	1	4	yes	—
226	#124	—	2	4	no	—
227	#124	—	2	4	no	—
228	#124	—	2	1	no	—
229	#124	C₂³×C₁₉	1	2	yes	—
230	#123	—	2	4	no	—
231	#123	—	2	4	no	—
232	C₂²⋊Q₈	—	2	4	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₂²⋊Q₈	—	2	2	no	—
235	C₂²⋊Q₈	—	2	8	no	—
236	#121	—	2	2	no	—
237	#121	—	2	2	no	—
238	C₄×Q₈	—	2	4	no	—
239	C₄×Q₈	—	2	2	no	—
240	C₄×Q₈	—	2	2	no	—
241	C₄×D₄	—	2	4	no	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	—	2	2	no	—
244	#117	—	2	2	no	—
245	#117	D₄×C₁₉	1	4	yes	—
246	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
247	#107	—	2	4	no	—
248	#106	—	2	4	no	—
249	C₄⋊C₈	—	2	2	no	—
250	#102	—	2	4	no	—
251	#102	—	2	4	no	—
252	C₂²⋊C₈	—	2	2	no	—
253	#97	—	2	1	no	—
254	#97	C₂³×C₁₉	1	4	yes	—
255	#97	C₂³×C₁₉	1	2	yes	—
256	C₁₉	#19054	1	1	yes	characteristic, direct product factor, Sylow
257	#41	—	4	4	no	—
258	#41	—	4	4	no	—
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	8	4	no	—
264	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	C₂×D₄×C₁₉	1	2	yes	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
276	Q₁₆	—	4	4	no	—
277	SD₁₆	—	4	4	no	—
278	SD₁₆	—	4	4	no	—
279	D₈	—	4	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	4	no	—
283	C₄⋊C₄	—	2	8	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
291	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
292	C₂²⋊C₄	—	2	4	no	—
293	C₂²⋊C₄	—	2	8	no	—
294	C₂²⋊C₄	—	2	4	no	—
295	C₂²⋊C₄	—	2	4	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	—	2	4	no	—
298	C₄²	—	2	4	no	—
299	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₁₉	1	2	yes	—
301	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
302	C₂³	—	2	2	no	—
303	C₂³	—	4	1	no	—
304	C₂³	C₁₉×#120	1	1	yes	characteristic
305	Q₈	—	4	2	no	—
306	Q₈	—	4	4	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	4	no	—
309	Q₈	—	8	4	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	4	no	—
314	D₄	—	8	2	no	—
315	D₄	—	8	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	C₁₉×#120	1	2	yes	—
326	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
328	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
329	C₈	—	2	4	no	—
330	C₈	—	4	4	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	8	2	no	—
337	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	4	no	—
343	C₄	—	4	4	no	—
344	C₄	—	8	1	no	—
345	C₄	C₁₉×#453	1	2	yes	—
346	C₂	—	4	1	no	—
347	C₂	—	8	2	no	—
348	C₂	—	16	1	no	—
349	C₂	C₂×C₁₉×#453	1	1	yes	characteristic
350	C₂	C₁₉×#3020	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₁₉×#3150	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₁₉×#3550	C₂	semidirect
2. C₁₉×#3235	C₂	semidirect
4. C₁₉×#1660	C₂	semidirect
5. C₁₉×#1657	C₂	semidirect
6. C₁₉×#1536	C₂	semidirect
7. C₁₉×#1531	C₂	semidirect
126. #19054	C₁₉	direct (N × H)
156. #3550	C₂×C₁₉	semidirect
157. #3235	C₂×C₁₉	semidirect
159. #1660	C₂×C₁₉	semidirect
160. #1657	C₂×C₁₉	semidirect
161. #1536	C₂×C₁₉	semidirect
162. #1531	C₂×C₁₉	semidirect
256. C₁₉	#19054	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₉×#3550	2
C₁₉×#3235	2
C₁₉×#3150	2
C₁₉×#1660	2
C₁₉×#1657	2
C₁₉×#1536	2
C₁₉×#1531	2
#19054	19

Subgroups — C19×#19054 #18025013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#19054 #18025013